A Theory Enthroned

Number theory is to mathematics as evolution is to biology or electromagnetism to physics: None is the final word in its discipline, but each is a fundamental and fascinating subject. The great German mathematician Carl Friedrich Gauss called number theory the "Queen of Mathematics." As Jay Goldman shows in this combination history and textbook, number theory's reign has lasted several hundred years by now—and its influence is only growing.

The roots of number theory go way back, at least as far as Pythagoras in the 5th or 6th century b.c. But the subject really got started through the efforts of one individual: Pierre de Fermat, the 17th-century French jurist, whose famous "Last Theorem" was only recently settled by Princeton University number theorist Andrew Wiles.

Fermat was the first to observe some of the subtle properties of primes. He noticed, for example, that any number raised to a prime power has itself as a remainder when divided by the prime. He also noticed that the primes 5, 13, 17 and so on—that is, those that are 1 more than a multiple of 4—can always be expressed as the sum of an even and an odd square (5 = 4 + 1, 13 = 4 + 9, 17 = 16 + 1 and so on). Fermat rarely wrote his proofs down, though; he customarily communicated his results through correspondence, as challenges to other mathematicians.

Fermat tried to interest his contemporaries in number theory but without much success. The subject really took off nearly a hundred years later, with the Swiss mathematician Leonhard Euler. Euler supplied the first published proofs for many of Fermat's results and extended number theory into new regions. In France, Joseph Louis Lagrange and Adrien-Marie Legendre also took up the cause. It was Gauss, though, who took number theory to a new level. His book, Disquisitiones Arithmeticae, written in his early 20s, gave shape to the subject and went way beyond what anyone else had done before.

After Gauss, things sped up. The 19th century saw the development of algebraic and analytic number theory, as well as important discoveries about "transcendental" numbers and the geometry of numbers. The subject has continued to accelerate throughout the 20th century and shows no signs of slowing down in the 21st.

Goldman surveys most of the major aspects of number theory. The one notable exception is analytic number theory: The Prime Number Theorem and its intimate relationship with the Riemann zeta function are barely even referred to. To give the analytic side its due, however, would have added at least another hundred pages to the book.

The historical highlights are especially welcome. They include excerpts from Fermat's correspondence and Gauss's Disquisitiones. Most number theory books merely present the theory as it's been refined over the years.

It's good to get a glimpse into the way the subject actually developed and the thinking of the people who made the key discoveries. Gauss's royal reference to number theory might anymore suggest an outmoded indulgence of a dissolute and effete social class (if not something worse). But it nonetheless retains the intended sense of majestic splendor, even to resolutely egalitarian ears. The theory of numbers, from Fermat to Wiles, is among the crown jewels of mathematics. As the younger generation would say today, number theory rules.—Barry Cipra, Northfield, Minnesota